Question

The ratio of athletes in the two teams is 3:2. After 12 athletes move from first team to second, the ratio is 5:4. Find the number of athletes on each team originally.

Answer 1

Answer:

1. 162 athletes
2. 108 athletes

Step-by-step explanation:

Each ratio can be expressed using two numbers that have the same sum. The sum in the first given ratio is 3+2 = 5, and in the second ratio, it is 5+4 = 9. The least common multiple of 5 and 9 is 45, so we can express both ratios so they have a sum of 45:

  3 : 2 = 3(9) : 2(9) = 27 : 18

  5 : 4 = 5(4) : 4(5) = 25 : 20

Here, we observe that the second ratio has numbers that look like 2 was moved from the numerator to the denominator. (27 -2 = 25; 18 +2 = 20) The problem statement tells us 12 athletes were moved, so multiplying the first ratio by 12/2 = 6 will tell us the numbers of players on each original team:

  first team = 27×6 = 162 athletes

  second team = 18×6 = 108 athletes

Answer 2

Answers:

• team one = 162
• team two = 108

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Explanation:

The ratio 3:2 scales up to 3x:2x for some positive whole number x.

This tells us that the first team has 3x people and the second team has 2x people. We'll subtract 12 from the first team and add 12 to the second team to represent 12 people moving.

• 3x becomes 3x-12
• 2x becomes 2x+12

The new team counts 3x-12 and 2x+12 represent team 1 and team 2. Those expressions are in the new proportion 5:4.

This means....

(team 1 new count)/(team 2 new count) = 5/4

(3x-12)/(2x+12) = 5/4

4(3x-12) = 5(2x+12)

12x-48 = 10x+60

12x-10x = 60+48

2x = 108

x = 108/2

x = 54

From this, we then can say:

• team 1 original count = 3x = 3*54 = 162
• team 2 original count = 2x = 2*54 = 108

The two subtotals add to 162+108 = 270 players overall.

If we were to subtract 12 from team 1, and add 12 to team 2, then,

• team 1 new count = 162-12 = 150
• team 2 new count = 108+12 = 120

The ratio 150:120 then reduces to 5:4 when we divide both parts by the GCF 30. This helps confirm we have the correct answer.